12. Consider the Merton capital structure model. (a) Show that the (real-world) recovery rate (as a fraction of the face value) on the bond is

of the form R = min (1, chm—6)) Where c is the negative of the (real-world) distance to default, b = cn/T and X =

ln(VT/Vg):/(ﬂ.—0'2/2)T

a T ' (b) Derive an expression for the expected recovery rate E [RI [X34] where I [X9] is the default

indicator. Hint: You might ﬁnd something on the ﬁnal exam sheet of “Potentially Useful

Facts” useful. (c) Show that the expected (log) return on equity converges to ,u as leverage goes to zero. ((1) Derive an expression for the (continuously—compounded) yield on the debt, which is the

value that solves D0 = e‘yTL, along With the credit spread y~— 7‘. Your expression for credit spread should depend only on 0V, T, and the leverage L = Law. (e) Assuming ,uv = 0.1, (IV = 0.2, L = 0.85, V}, = 1, r = 0.02, and T = 4, compute the

real-world probability of default.